3.1632 \(\int \frac{x^3}{\left (a+\frac{b}{x}\right )^3} \, dx\)

Optimal. Leaf size=86 \[ -\frac{b^6}{2 a^7 (a x+b)^2}+\frac{6 b^5}{a^7 (a x+b)}+\frac{15 b^4 \log (a x+b)}{a^7}-\frac{10 b^3 x}{a^6}+\frac{3 b^2 x^2}{a^5}-\frac{b x^3}{a^4}+\frac{x^4}{4 a^3} \]

[Out]

(-10*b^3*x)/a^6 + (3*b^2*x^2)/a^5 - (b*x^3)/a^4 + x^4/(4*a^3) - b^6/(2*a^7*(b +
a*x)^2) + (6*b^5)/(a^7*(b + a*x)) + (15*b^4*Log[b + a*x])/a^7

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Rubi [A]  time = 0.145027, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{b^6}{2 a^7 (a x+b)^2}+\frac{6 b^5}{a^7 (a x+b)}+\frac{15 b^4 \log (a x+b)}{a^7}-\frac{10 b^3 x}{a^6}+\frac{3 b^2 x^2}{a^5}-\frac{b x^3}{a^4}+\frac{x^4}{4 a^3} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b/x)^3,x]

[Out]

(-10*b^3*x)/a^6 + (3*b^2*x^2)/a^5 - (b*x^3)/a^4 + x^4/(4*a^3) - b^6/(2*a^7*(b +
a*x)^2) + (6*b^5)/(a^7*(b + a*x)) + (15*b^4*Log[b + a*x])/a^7

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{4}}{4 a^{3}} - \frac{b x^{3}}{a^{4}} + \frac{6 b^{2} \int x\, dx}{a^{5}} - \frac{10 b^{3} x}{a^{6}} - \frac{b^{6}}{2 a^{7} \left (a x + b\right )^{2}} + \frac{6 b^{5}}{a^{7} \left (a x + b\right )} + \frac{15 b^{4} \log{\left (a x + b \right )}}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(a+b/x)**3,x)

[Out]

x**4/(4*a**3) - b*x**3/a**4 + 6*b**2*Integral(x, x)/a**5 - 10*b**3*x/a**6 - b**6
/(2*a**7*(a*x + b)**2) + 6*b**5/(a**7*(a*x + b)) + 15*b**4*log(a*x + b)/a**7

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Mathematica [A]  time = 0.0900394, size = 73, normalized size = 0.85 \[ \frac{a^4 x^4-4 a^3 b x^3+12 a^2 b^2 x^2+\frac{2 b^5 (12 a x+11 b)}{(a x+b)^2}+60 b^4 \log (a x+b)-40 a b^3 x}{4 a^7} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b/x)^3,x]

[Out]

(-40*a*b^3*x + 12*a^2*b^2*x^2 - 4*a^3*b*x^3 + a^4*x^4 + (2*b^5*(11*b + 12*a*x))/
(b + a*x)^2 + 60*b^4*Log[b + a*x])/(4*a^7)

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Maple [A]  time = 0.012, size = 83, normalized size = 1. \[ -10\,{\frac{{b}^{3}x}{{a}^{6}}}+3\,{\frac{{b}^{2}{x}^{2}}{{a}^{5}}}-{\frac{b{x}^{3}}{{a}^{4}}}+{\frac{{x}^{4}}{4\,{a}^{3}}}-{\frac{{b}^{6}}{2\,{a}^{7} \left ( ax+b \right ) ^{2}}}+6\,{\frac{{b}^{5}}{{a}^{7} \left ( ax+b \right ) }}+15\,{\frac{{b}^{4}\ln \left ( ax+b \right ) }{{a}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b/x)^3,x)

[Out]

-10*b^3*x/a^6+3*b^2*x^2/a^5-b*x^3/a^4+1/4*x^4/a^3-1/2*b^6/a^7/(a*x+b)^2+6*b^5/a^
7/(a*x+b)+15*b^4*ln(a*x+b)/a^7

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Maxima [A]  time = 1.46691, size = 123, normalized size = 1.43 \[ \frac{12 \, a b^{5} x + 11 \, b^{6}}{2 \,{\left (a^{9} x^{2} + 2 \, a^{8} b x + a^{7} b^{2}\right )}} + \frac{15 \, b^{4} \log \left (a x + b\right )}{a^{7}} + \frac{a^{3} x^{4} - 4 \, a^{2} b x^{3} + 12 \, a b^{2} x^{2} - 40 \, b^{3} x}{4 \, a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(a + b/x)^3,x, algorithm="maxima")

[Out]

1/2*(12*a*b^5*x + 11*b^6)/(a^9*x^2 + 2*a^8*b*x + a^7*b^2) + 15*b^4*log(a*x + b)/
a^7 + 1/4*(a^3*x^4 - 4*a^2*b*x^3 + 12*a*b^2*x^2 - 40*b^3*x)/a^6

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Fricas [A]  time = 0.217278, size = 158, normalized size = 1.84 \[ \frac{a^{6} x^{6} - 2 \, a^{5} b x^{5} + 5 \, a^{4} b^{2} x^{4} - 20 \, a^{3} b^{3} x^{3} - 68 \, a^{2} b^{4} x^{2} - 16 \, a b^{5} x + 22 \, b^{6} + 60 \,{\left (a^{2} b^{4} x^{2} + 2 \, a b^{5} x + b^{6}\right )} \log \left (a x + b\right )}{4 \,{\left (a^{9} x^{2} + 2 \, a^{8} b x + a^{7} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(a + b/x)^3,x, algorithm="fricas")

[Out]

1/4*(a^6*x^6 - 2*a^5*b*x^5 + 5*a^4*b^2*x^4 - 20*a^3*b^3*x^3 - 68*a^2*b^4*x^2 - 1
6*a*b^5*x + 22*b^6 + 60*(a^2*b^4*x^2 + 2*a*b^5*x + b^6)*log(a*x + b))/(a^9*x^2 +
2*a^8*b*x + a^7*b^2)

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Sympy [A]  time = 1.82054, size = 92, normalized size = 1.07 \[ \frac{12 a b^{5} x + 11 b^{6}}{2 a^{9} x^{2} + 4 a^{8} b x + 2 a^{7} b^{2}} + \frac{x^{4}}{4 a^{3}} - \frac{b x^{3}}{a^{4}} + \frac{3 b^{2} x^{2}}{a^{5}} - \frac{10 b^{3} x}{a^{6}} + \frac{15 b^{4} \log{\left (a x + b \right )}}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(a+b/x)**3,x)

[Out]

(12*a*b**5*x + 11*b**6)/(2*a**9*x**2 + 4*a**8*b*x + 2*a**7*b**2) + x**4/(4*a**3)
- b*x**3/a**4 + 3*b**2*x**2/a**5 - 10*b**3*x/a**6 + 15*b**4*log(a*x + b)/a**7

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GIAC/XCAS [A]  time = 0.22425, size = 112, normalized size = 1.3 \[ \frac{15 \, b^{4}{\rm ln}\left ({\left | a x + b \right |}\right )}{a^{7}} + \frac{12 \, a b^{5} x + 11 \, b^{6}}{2 \,{\left (a x + b\right )}^{2} a^{7}} + \frac{a^{9} x^{4} - 4 \, a^{8} b x^{3} + 12 \, a^{7} b^{2} x^{2} - 40 \, a^{6} b^{3} x}{4 \, a^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(a + b/x)^3,x, algorithm="giac")

[Out]

15*b^4*ln(abs(a*x + b))/a^7 + 1/2*(12*a*b^5*x + 11*b^6)/((a*x + b)^2*a^7) + 1/4*
(a^9*x^4 - 4*a^8*b*x^3 + 12*a^7*b^2*x^2 - 40*a^6*b^3*x)/a^12